Or was this term in use before he used it in a song? I had never heard it before his song, but it seems to have caught on as a minor cliché.
I can’t cite anything, but I know I heard the term when I was a child, loong before Paul Simon was born.
The phrase “one trick pony” has been in use for decades, long long long before you were a twinkle in your father’s eye. It means a person who can do one thing well, like an actor can do a great ‘rage’ scene, but is a flop at romance or comedy. No, I don’t have a cite. No, Paul Simon did not invent it. Why would you think he did?
From dictionary.com
Main Entry: one-trick pony
Part of Speech: n
Definition: any person or group with only one discernable trait, talent, or area of expertise
Example: That actor proved to be a one-trick pony.
Etymology: from act in Cuffling Cousins Circus of mid-1800s
I’m pretty sure I’d heard it before Simon used it. I figured it referred to a rodeo bucking bronco that had a single, often effective, move for dislodging a rider. But there’s no evidence for that interpretation that I know.
Bridging people together is my job.
Thank you 
Reading the OP gives people less opporunity to be snarky…
I was about 23 when the album came out.
Likewise and likewise.
This is correct for the pony who only knew one trick. In a literal sense.
As to the figurative sense–the OED only shows it from 1950.